Comments for MEDB 5501, Week 13

Covariance

\(Cov(X,Y)=\frac{1}{n-1}\Sigma(X_i-\bar{X})(Y_i-\bar{Y})\)
- \((X_i-\bar{X})(Y_i-\bar{Y})\) is positive if
  - \(X_i\) and \(Y_i\) both above average
  - \(X_i\) and \(Y_i\) both below average
- \((X_i-\bar{X})(Y_i-\bar{Y})\) is negative if
  - \(X_i\) above average and \(Y_i\) below average
  - \(X_i\) below average and \(Y_i\) above average

I want to start this section with a discussion of covariance. Covariance is a term that the more mathematically oriented statisticians love to use. It is an interesting statistic from a theoretical perspective and it forms the foundation for a large number of statistical tests.

It doesn’t, however, have as much practical application, compared to the correlation, which you will see in just a bit. I am introducing it here to get you familiar with the terminology.

It is sort of analogous to the term variance versus standard deviation. The variance is interesting from a theoretical perspective, but the standard deviation has far more practical implications.

To compute the covariance, you add up a bunch of terms, each of which is a product. The product is positive if both X and Y are above average or if both X and Y are below average. A positive times a positive is positive and a negative times a negative is also positive.

The product is negative if one value is above average and the other value is below average. A positive times a negative or a negative times a positive produces a negative product.

Think of the covariance as measuring the tendency for two variables to co-vary in a positive sense (large pairing with large and small pairing with small) or in a negative sense (large pairing with small and small pairing with large).

\(\ \)

\(\bar{X}=20\);

\(\bar{Y}=10\);

\(S_X=6.5\);

\(S_Y=3.7\)

Calculation of covariance

 x_centered y_centered product
          9         -5     -45
          1         -3      -3
         -5         -1       5
          5          1       5
         -9          3     -27
         -1          5      -5

\(\ \)

\(Cov(X,Y)=\frac{1}{5}(-70)=-14\)

Correlation

\(Corr(X,Y)=\frac{Cov(X,Y)}{S_XS_Y}\)
- Also use \(r_{XY}\)
- Population correlation is \(\rho_{XY}\)

Calculation of correlation

\(r_{XY}=\frac{-14}{6.5 \times 3.7}=-0.571929\)
- Always round!
  - \(r_{XY}=-0.57\) or \(-0.6\)

Interpretation of correlation

r is always between -1 and +1
- Positive values imply positive association
- Negative values imply negative association
- Strongest associations closest to -1 or +1

r between -1 and -0.7, strong negative association

r between -0.7 and -0.3, weak negative association

r between -0.3 and +0.3, little or no association

r between +0.3 and +0.7, weak positive association

r between +0.7 and +1, strong positive association

Extreme case, perfect association

Sleep data dictionary, 1 of 6

---
data_dictionary:
  sleep.txt
  
source:
  This dataset is part of the Austrasian Data and
  Story Library (OZDASL). Please cite this data as
  Smyth, GK (2011). Australasian Data and Story 
  Library (OzDASL). http://www.statsci.org/data.
  The data comes originally from Allison, T., and
  Cicchetti, D. V. (1976). Sleep in mammals. 
  ecological and constitutional correlates. 
  Science 194 (November 12), 732-734.

Sleep data dictionary, 2 of 6

description:
  This dataset has information about sleep patterns
  in 62 common mammals, along with other information
  that might help you understand what influences
  variations in sleep.
  
download:
  text-format: http://www.statsci.org/data/general/sleep.txt
  additional-information: http://www.statsci.org/data/general/sleep.html

copyright:
  There is no information about the copyright for this
  dataset. You should, however, be able to use this
  data for individual educational purposes under the
  Fair Use guidelines of U.S. copyright law.

Sleep data dictionary, 3 of 6

format: 
  delimiter: tab
  varnames: included in the first row of data
  missing-value-code: NA
  rows: 62
  columns: 11

Sleep data dictionary, 4 of 6

vars:
  Species:
    label: Species of mammal
    
  BodyWt:
    label: Body weight
    unit: kg
    
  BrainWt:
    label: Brain weight
    unit: g

Sleep data dictionary, 5 of 6

  NonDreaming:
    label: Time spent in non-dreaming sleep
    unit: hours
    
  Dreaming:
    label: Time spent in dreaming sleep
    unit: hours
    
  TotalSleep:
    label: Total time spent in sleep
    unit: hours
    
  LifeSpan:
    unit: years

Sleep data dictionary, 6 of 6

  Gestation:
    unit: days

  Predation:
    scale: likert
    range: 1-5

  Exposure:
    scale: likert
    range: 1-5
    
  Danger:
    scale: likert
    range: 1-5
---

What does a missing value represent

Dropout
Refuse to answer survey question
Survey question is not applicable
Lab result is lost
Concentration below detectable limit
Many other reasons

Common missing value codes

A single dot (.)
- SPSS and SAS
NA
- R
Asterisk (*) and other symbols
Unusual number codes (-1, 9, 99, 999)

Importing missing values

No problems for default value
NA and * convert numeric to string
- Fix during import, or
- Convert back after import
Unusual number codes
- Designate after import
- Don’t forget!

When you are importing a dataset with missing data into SPSS, use of the default code, a single dot, will usually work just fine.

The problem occurs when the data you get uses a different code, like NA or asterisk. SPSS will often take a column of numbers with one or more missing value codes, and convert it into a string. This makes it impossible for you to run most of the analyses that you would want to run.

You can tell SPSS during the import to designate the column of data as numeric, and SPSS will automatically convert any NA or asterisk to missing. Or you can convert from string to numeric after import. The conversion process will convert your NA and asterisk to missing.

If your dataset uses number codes for missing, you should have no trouble during import, but you do need to designate which numeric code or codes represents a missing value.

Don’t forget this, or any statistics that you compute will be wrong, sometimes very wrong.

Imputing missing values, 1 of 2

Several simple (simplistic?) imputation choices
- No news is bad news
- No news is good news
- No news is average news (MCAR)
- No news is last week’s news (LOCF)

Sometimes you can use a bit of knowledge about the context of your research to help infer what the missing value might be.

A common scenario is what I call “no news is bad news”. If someone drops out of a weight loss study, there’s a good chance it is because the intervention was not effective. You might make a similar assumption for a smoking cessation study. Now you might consider this a bit extreme. But it is a more realistic scenario than some of the other choices that you might make.

You might also consider the opposite scenario, “no news is good news”. If you have information about adverse events in a drug trial, perhaps it is because people forget to say “no problems” more often than they forget to specify problems.

A third scenario is “no news is average news”. This might apply when the reason an observation is missing stems from a cause that is totally unrelated to anything else in the study. You might not have results from a blood test because a technician lost one of your blood samples. That would not be related to any treatment received or any outcome that could have been measured. It was just dumb luck. In such a case, you might replace the missing value with the average of the non-missing values. This case is often called missing completely at random, known by the acronym MCAR.

A final scenario is “no news is last week’s news”. If you are measuring a patient longitudinally, and the patient misses the last visit, you might take the value from the next to last visit and assume that things have not changed too much from the previous week (or month). This is often referred to as last observation carried forward, known by the acronym LOCF.

Imputing missing values, 2 of 2

Rigorous approaches (beyond the scope of this class)
- Missing at random (MAR), Missing not at random (MNAR)
- Ignorable, Non-ignorable
- Single/Multiple imputation
- Maximum likelihood/Bayesian approaches
You cannot ignore missingness, you cannot avoid imputation

Statisticians will sometimes classify missingness into two other categories besides missing completely at random. There is missing at random, where missingness might depend on some covariates like age and gender. This might allow you to predict missingness using some type of regression model.

If you use regression models to impute the missing values, you need to do it carefully. The regression model itself has some uncertainty associated with it. Failure to account for this uncertainty can lead to falsely precise results. Generally, you need to impute the missing values multiple times (multiple imputation), but practical considerations may force you to impute just once (single imputation).

In contrast, if the data is described as missing not at random, then missingness might be related to the missing value itself. The cases of “no news is bad news” and “no news is good news” are extreme examples of this. Generally, things get messy if a missing outcome is related to what the outcome might have been if you could have observed the value. You often have to make untestable assumptions about your data.

In addition to the regression approaches, there are methods based on maximum likelihood principles or Bayesian models. It turns out that Bayesian models are quite good at handling the messiest case, the missing not at random case.

While I will not expect you to apply these complex approaches, I did want to make you aware of some of the terminology used when discussing missing values.

One point that bears remembering is that you cannot ignore missing values. If you try, you are effectively adopting a “no news is average news” assumption.

It is somewhat analogous to the saying “not to decide is to decide.” Ignoring missingness is effectively imputing a value for missing data that is averaging out the missing data. So you will end up imputing anyway.

SPSS investigation of missing data, 1 of 2

SPSS has a nice set of procedures that help you investigate patterns in missing values that may help you understand the processes behind missingness. This might guide you towards an appropriate method of imputation. Here is one table from SPSS that shows how often values are missing and also tries to identify extreme values. These might be missing value codes that you forget to tell SPSS about, or they might represent values that you have to designate as missing because they are so extreme that they cannot be anything other than a coding error. Examples might be a body mass index of 3.2, which is incompatible with life.

Converting an extreme value to a missing value is something that you should not do with caution and only after discussing this choice with your research team. Sometimes the outlier itself may be the only interesting feature of your data and you don’t want to toss it aside without thinking in depth about it.

SPSS investiation of missing data, 2 of 2

Missing value approaches for correlations, 1 of 2

\[\begin{matrix} A_1 & B_1 & C_1\\ A_2 & B_2 & C_2\\ A_3 & B_3 & C_3\\ A_4 & B_4 & C_4\\ A_5 & B_5 & C_5\\ A_6 & B_6 & .\\ \end{matrix}\]

Missing value approaches for correlations, 2 of 2

Complete case analysis,
- Use 9 pairs for \(r_{AB}\), \(r_{AC}\), and \(r_{BC}\)
Pairwise deletion
- Use 9 pairs for \(r_{AC}\), and \(r_{BC}\)
- Use 10 pairs for \(r_{AB}\)

SPSS correlation analysis, 1 of 6

SPSS analysis, 4 of 9

SPSS analysis, 5 of 9

SPSS analysis, 6 of 9

SPSS analysis, 7 of 9

SPSS analysis, 8 of 9

SPSS analysis, 9 of 9

SPSS analysis, 10 of 9

SPSS analysis, 11 of 9

Partial correlation

\(\rho_{XY\cdot Z}=\frac{\rho_{XY}-\rho_{XZ}\rho_{ZY}} {\sqrt{1-\rho_{XZ}^2}\sqrt{1-\rho_{ZY}^2}}\)